Problem: Simplify; express your answer in exponential form. Assume $r\neq 0, t\neq 0$. $\dfrac{{(r^{-2}t^{-3})^{4}}}{{(r^{-4}t^{-3})^{-3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(r^{-2}t^{-3})^{4} = (r^{-2})^{4}(t^{-3})^{4}}$ On the left, we have ${r^{-2}}$ to the exponent ${4}$ . Now ${-2 \times 4 = -8}$ , so ${(r^{-2})^{4} = r^{-8}}$ Apply the ideas above to simplify the equation. $\dfrac{{(r^{-2}t^{-3})^{4}}}{{(r^{-4}t^{-3})^{-3}}} = \dfrac{{r^{-8}t^{-12}}}{{r^{12}t^{9}}}$ Break up the equation by variable and simplify. $\dfrac{{r^{-8}t^{-12}}}{{r^{12}t^{9}}} = \dfrac{{r^{-8}}}{{r^{12}}} \cdot \dfrac{{t^{-12}}}{{t^{9}}} = r^{{-8} - {12}} \cdot t^{{-12} - {9}} = r^{-20}t^{-21}$